Theorem nb3grprlem1 29307

Description: Lemma 1 for nb3grpr 29309. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)

Hypotheses

Ref	Expression
nb3grpr.v	⊢ 𝑉 = (Vtx‘𝐺)
nb3grpr.e	⊢ 𝐸 = (Edg‘𝐺)
nb3grpr.g	⊢ (𝜑 → 𝐺 ∈ USGraph)
nb3grpr.t	⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
nb3grpr.s	⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))

Assertion

Ref	Expression
nb3grprlem1	⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

Proof of Theorem nb3grprlem1

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nb3grpr.s	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
2		prid1g 4724	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ {𝐵, 𝐶})
3	2	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐵 ∈ {𝐵, 𝐶})
4	1, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶})
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ {𝐵, 𝐶})
6		eleq2 2817	. . . . . . 7 ⊢ ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
7	6	eqcoms 2737	. . . . . 6 ⊢ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
8	7	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
9	5, 8	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ (𝐺 NeighbVtx 𝐴))
10		nb3grpr.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ USGraph)
11		nb3grpr.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
12	11	nbusgreledg 29280	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐵, 𝐴} ∈ 𝐸))
13		prcom 4696	. . . . . . . . 9 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
14	13	a1i 11	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → {𝐵, 𝐴} = {𝐴, 𝐵})
15	14	eleq1d 2813	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ({𝐵, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
16	12, 15	bitrd 279	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
17	10, 16	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
19	9, 18	mpbid 232	. . 3 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ 𝐸)
20		prid2g 4725	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐵, 𝐶})
21	20	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ {𝐵, 𝐶})
22	1, 21	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ {𝐵, 𝐶})
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ {𝐵, 𝐶})
24		eleq2 2817	. . . . . . 7 ⊢ ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
25	24	eqcoms 2737	. . . . . 6 ⊢ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
26	25	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
27	23, 26	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
28	11	nbusgreledg 29280	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
29		prcom 4696	. . . . . . . . 9 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
30	29	a1i 11	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → {𝐶, 𝐴} = {𝐴, 𝐶})
31	30	eleq1d 2813	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
32	28, 31	bitrd 279	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
33	10, 32	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
34	33	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
35	27, 34	mpbid 232	. . 3 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐶} ∈ 𝐸)
36	19, 35	jca 511	. 2 ⊢ ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
37		nb3grpr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
38	37, 11	nbusgr 29276	. . . . 5 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
39	10, 38	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 NeighbVtx 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
40	39	adantr 480	. . 3 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
41		nb3grpr.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 = {𝐴, 𝐵, 𝐶})
42		eleq2 2817	. . . . . . . . . 10 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
43	41, 42	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
44	43	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
45		vex 3451	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
46	45	eltp 4653	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴 ∨ 𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
47	11	usgredgne 29133	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → 𝐴 ≠ 𝑣)
48		df-ne 2926	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ≠ 𝑣 ↔ ¬ 𝐴 = 𝑣)
49		pm2.24 124	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
50	49	eqcoms 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
51	50	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
52	48, 51	sylbi 217	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ≠ 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
53	47, 52	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
54	53	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USGraph → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
55	10, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
56	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
57	56	com3r 87	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
58		orc 867	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
59	58	2a1d 26	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
60		olc 868	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐶 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
61	60	2a1d 26	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
62	57, 59, 61	3jaoi 1430	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐴 ∨ 𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
63	46, 62	sylbi 217	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
64	63	com12 32	. . . . . . . 8 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
65	44, 64	sylbid 240	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
66	65	impd 410	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
67		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = 𝐵
68	67	3mix2i 1335	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)
69	1	simp2d 1143	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
70		eltpg 4650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)))
71	69, 70	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)))
72	68, 71	mpbiri 258	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
73	72	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
74		eleq1 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
75	74	bicomd 223	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝐵 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
76	75	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
77	73, 76	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
78	42	bicomd 223	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
79	41, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
80	79	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
81	77, 80	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 = 𝐵) → 𝑣 ∈ 𝑉)
82	81	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 = 𝐵 → 𝑣 ∈ 𝑉))
83	82	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → 𝑣 ∈ 𝑉))
84	83	impcom 407	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣 ∈ 𝑉)
85		preq2 4698	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
86	85	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
87	86	eqcoms 2737	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
88	87	biimpcd 249	. . . . . . . . . . . 12 ⊢ ({𝐴, 𝐵} ∈ 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
89	88	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
90	89	impcom 407	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
91	84, 90	jca 511	. . . . . . . . 9 ⊢ ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
92	91	ex 412	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
93		tpid3g 4736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
94	93	3ad2ant3 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
95	1, 94	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
96	95	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
97		eleq1 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
98	97	bicomd 223	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝐶 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
99	98	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
100	96, 99	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
101	79	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
102	100, 101	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 = 𝐶) → 𝑣 ∈ 𝑉)
103	102	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 = 𝐶 → 𝑣 ∈ 𝑉))
104	103	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → 𝑣 ∈ 𝑉))
105	104	impcom 407	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣 ∈ 𝑉)
106		preq2 4698	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
107	106	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
108	107	eqcoms 2737	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
109	108	biimpcd 249	. . . . . . . . . . . 12 ⊢ ({𝐴, 𝐶} ∈ 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
110	109	ad2antll 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
111	110	impcom 407	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
112	105, 111	jca 511	. . . . . . . . 9 ⊢ ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
113	112	ex 412	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
114	92, 113	jaoi 857	. . . . . . 7 ⊢ ((𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
115	114	com12 32	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
116	66, 115	impbid 212	. . . . 5 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) ↔ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
117	116	abbidv 2795	. . . 4 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∣ (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)})
118		df-rab 3406	. . . 4 ⊢ {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝑣 ∣ (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)}
119		dfpr2 4610	. . . 4 ⊢ {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)}
120	117, 118, 119	3eqtr4g 2789	. . 3 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝐵, 𝐶})
121	40, 120	eqtrd 2764	. 2 ⊢ ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
122	36, 121	impbida 800	1 ⊢ (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707   ≠ wne 2925  {crab 3405  {cpr 4591  {ctp 4593  ‘cfv 6511  (class class class)co 7387  Vtxcvtx 28923  Edgcedg 28974  USGraphcusgr 29076   NeighbVtx cnbgr 29259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-fz 13469  df-hash 14296  df-edg 28975  df-upgr 29009  df-umgr 29010  df-usgr 29078  df-nbgr 29260

This theorem is referenced by:  nb3grpr  29309  nb3grpr2  29310  nb3gr2nb  29311